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Fractional Calculus and Fractional Processes with Applications to
Financial Economics presents the theory and application of
fractional calculus and fractional processes to financial data.
Fractional calculus dates back to 1695 when Gottfried Wilhelm
Leibniz first suggested the possibility of fractional derivatives.
Research on fractional calculus started in full earnest in the
second half of the twentieth century. The fractional paradigm
applies not only to calculus, but also to stochastic processes,
used in many applications in financial economics such as modelling
volatility, interest rates, and modelling high-frequency data. The
key features of fractional processes that make them interesting are
long-range memory, path-dependence, non-Markovian properties,
self-similarity, fractal paths, and anomalous diffusion behaviour.
In this book, the authors discuss how fractional calculus and
fractional processes are used in financial modelling and finance
economic theory. It provides a practical guide that can be useful
for students, researchers, and quantitative asset and risk managers
interested in applying fractional calculus and fractional processes
to asset pricing, financial time-series analysis, stochastic
volatility modelling, and portfolio optimization.
The study of heavy-tailed distributions allows researchers to
represent phenomena that occasionally exhibit very large deviations
from the mean. The dynamics underlying these phenomena is an
interesting theoretical subject, but the study of their statistical
properties is in itself a very useful endeavor from the point of
view of managing assets and controlling risk. In this book, the
authors are primarily concerned with the statistical properties of
heavy-tailed distributions and with the processes that exhibit
jumps. A detailed overview with a Matlab implementation of
heavy-tailed models applied in asset management and risk
managements is presented. The book is not intended as a theoretical
treatise on probability or statistics, but as a tool to understand
the main concepts regarding heavy-tailed random variables and
processes as applied to real-world applications in finance.
Accordingly, the authors review approaches and methodologies whose
realization will be useful for developing new methods for
forecasting of financial variables where extreme events are not
treated as anomalies, but as intrinsic parts of the economic
process.
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